Szemerédi-Trotter theorem

The number of incidences of a set of n points and a set of m lines in the real plane ℝ2 is

I=O⁢(n+m+(n⁢m)23).

Proof. Let’s consider the points as vertices of a graph, and connect two vertices by an edge if they are adjacent on some line. Then the number of edges is e=I-m. If e<4⁢n then we are done. If e≥4⁢n then by crossing lemma

m2≥cr⁡(G)≥164⁢(I-m)3n2,

and the theorem follows.

Recently, Tóth[1] extended the theorem to the complex planeℂ2. The proof is difficult.